From Wikipedia: (with my emphasis)
Regarding the aspect, there are two types of quotation marks:
'…' and "…", are known as neutral, vertical, straight, typewriter, "dumb", or "ASCII" quotation marks. The left and right marks are identical. These are found on typical English typewriters and computer keyboards, although they are sometimes automatically converted to the other type by software.
‘…’ and “…”, are known as typographic, curly, curved, book, or "smart" quotation marks. The beginning marks are commas raised to the top of the line and rotated 180°. The ending marks are commas raised to the top of the line. Curved quotation marks are used mainly in manuscript, printing and typesetting. Type cases (of any language) always have the correct quotation marks metal types for the respective language and never the vertical quotation marks metal types.
EDIT: It just occurred to me that rather than asking for the name of each type, you might be asking about the name of the property, which includes both options. If so, you could extend the descriptors used above to have "neutrality", "verticality", "straightness" or "dumbness" perhaps.